1508 OPTICS LETTERS / Vol. 21, No. 18 / September 15, 1996

Analysis of gratings with large periods and small featuresizes by stitching of the electromagnetic field

Ben Layet and Mohammad R. Taghizadeh

Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received February 12, 1996

We present a new method for the analysis of diffractive optical elements, which we refer to as field stitching.It is suitable for use with grating structures of arbitrarily large period, even when the local feature size is ofthe order of a wavelength. Furthermore, the concept is straightforwardly extendable to aperiodic structures.To assess its applicability, we have calculated the diffracted orders from a 1 3 81 fan-out grating with periodsof 100l and 10, 000l. The field-stitched calculations agree very well with independent rigorous predictions forthe small-period element and scalar-regime predictions for the large-period element. We believe that a varietyof areas within the diffractive-optics field will benefit from this new analytical tool. It promises accurateanalysis and, by facilitating component optimization, high-performance designs. 1996 Optical Society ofAmerica

One of the greatest stumbling blocks in the analysis ofdiffraction gratings is the combination of large periodand small feature size. Scalar optics is quite sufficientfor evaluating large-period structures as long as thefeatures are larger than , 10 wavelengths of theincident electromagnetic radiation. There also existmany rigorous theories (e.g., Refs. 1–3) that are usefulin the resonance regime, in which the feature sizes arecomparable with the size of the wavelength. However,the computational costs of rigorous methods rise toorapidly with the size of the problem (e.g., the period)for such methods to be feasible for many gratings withthe computing power available to most researchers.The subject of this Letter is a simple method thatpermits analysis of extremely large-period gratings,including those that contain very fine structural detail.For example, a grating with a period of thousandsof wavelengths and wavelength size features can bestudied. The method allows one to determine thedegree of convergence to the exact answer (as normalrigorous methods do), but it produces accurate resultswithout requiring excessive computing power.

The new method is essentially a two-stage process,although it is implemented with the two stagesmixed together for computational efficiency. First,the grating is considered to be divided into overlap-ping sections. A rigorous grating method is used tosolve the diffraction problem for each section as if itwere the whole period of a smaller grating. If thesection size, or local period, is large enough, thenthe calculated field near the middle of the section issimilar to the f ield that would be found there by useof a full rigorous calculation of the whole grating.Therefore we can use the central field from eachgrating section to get a piecewise representation ofthe field across the whole grating. At the surface ofthe grating region we match this representation to aRayleigh expansion containing all the nonevanescentorders to f ind the propagating f ield. We term themethod field stitching because of the piecewise evalu-ation of the f ield. To our knowledge this approach

0146-9592/96/181508-03$10.00/0

was not previously used in the study of diffractiongratings. The mathematical details are given below.

The derivation is presented for the transmittedfield. It does not depend on the polarization of theincident radiation or on the nature of grating refractiveindices. These are taken care of by the local rigorouscalculation, which we treat as a black box. The fieldUt outside the grating region is described by theRayleigh expansion

Ut ­p­X

p­2`

Tp expnifgpx 2 tpsz 2 hdg

o, (1)

where Tp are the amplitudes of the diffracted ordersand the propagation constants in the axial directionsare given by gp ­ g0 1 2ppyd, g0 ­ n0k sin u, andtp ­ fsn0kd2 2 gp

2g1/2; d is the total period, n0 is therefractive index of the incidence medium, u is theangle of the incident radiation, h is the grating regionthickness, and k ­ 2pyl is the free-space wavelength.

The local diffraction calculations for the gratingsections are made at regular intervals along thex axis. The sections are numbered from 1 to n, andthe parameters associated with the nth section areshown in Fig. 1. On the boundary of the gratingregion, z ­ h, we describe the f ield Ub, using theRayleigh expansions determined by these local calcu-lations:

Ub ­m­X

m­2`

T sndm dn expfigm

0sx 1 xsdg,

sn 2 1dw # x # nw , (2)

where T sndm are the diffracted-order amplitudes of the

nth grating section, gm0 ­ g0 1 2pmydl, dl is the local

period, w is the size of the central segment used fromeach local calculation, xs is the distance from the startof each local period to the central utilized segment, andthe factor dn ­ expfikn0 sinsud sn 2 2dwg arises fromthe variation in the phase of the incident plane waveacross the grating.

1996 Optical Society of America

8662(k )

September 15, 1996 / Vol. 21, No. 18 / OPTICS LETTERS 1509

Fig. 1. Schematic representation of a grating and asso-ciated parameters.

On the line z ­ h the total grating Rayleigh expan-sion Ut can be equated with Ub. The z-dependenceterm disappears from the exponential in the expres-sion for Ut. Furthermore, we can treat Ut as a Fourierexpansion of Ub and, by considering Tp the Fourier co-efficients, obtain the following expression for the trans-mitted order amplitudes:

Tp ­1d

NXn­1

ms1dXm­ms2d

T sndm dn exphigm

0fxs 1 s1 2 ndwgj

3Z nw

sn21dwexpfigm0xg exps2igpxd dx . (3)

The integral can be easily evaluated analytically toyield an expression for Tp that is a simple doublesum. The choice of the range over which the innersum should be calculated requires discussion. Con-sider the diffracted orders Tp. They vary with spatialfrequencies fp ­ gpy2p along the line z ­ h. Simi-larly, the locally calculated orders Tm from which Tpare found vary with spatial frequencies fm

0 ­ gm0y2p.

Now, if fm0 .. fp

0, then the mth term in Eq. (3) willnot produce a significant contribution to the ampli-tude of the pth diffracted order. The integral willevaluate to almost 0. So deciding the largest valueof interest of the order p places an upper limit onthe number of terms retained in the inner sum ofthe equation. Frequently one will be concerned withonly the propagating diffracted orders of the grating.It should be recalled that, regardless of the gratingperiod, these orders cannot vary more rapidly thanthe wavelength (or they would be evanescent). Thissuggests the inclusion in the sum of all local propagat-ing orders (that have the same maximum spatial fre-quency) plus enough evanescent orders to ensure thatno sizable contributions to the Tp are ignored. In factwe choose no evanescent orders because we present noresults for very high-angle orders. In this case,

ms1d ­ largest integer #dl

lfnt 2 n0 sinsudg ,

ms2d ­ smallest integer $dl

lf2nt 2 n0 sinsudg , (4)

where nt is the refractive index of the transmissionmedium. We have observed that, even with suchlimits on the inner sum, high-angle orders are sur-prisingly well predicted. Finally, we should make itclear that the above discussion does not imply the

exclusion of evanescent orders from the underlyingrigorous calculation. Their presence is necessary be-cause they cause a redistribution of energy that affectsthe amplitudes of the propagating orders.

To illustrate the performance of the field-stitchingmethod, a binary 1 3 81 fan-out grating with a sub-strate refractive index of 1.45 is analyzed under normalTE-polarized illumination for two different periods.We designed the grating, using paraxial scalar theory,for operation in the far f ield, so we expect to see a per-formance degradation for small periods. In all caseswe field stitch with a local period of dl ­ 50l and usethe rigorous Burckardt–Kaspar–Knop (BKK) method4

to do the local calculation. The central segment size wand the number of orders used in the BKK calculation,l, depend on the number of grating features in the localperiod. We can verify the diffraction pattern when theperiod is suff iciently small, using a full rigorous analy-sis, and when it is large a scalar diffraction analysis isappropriate. Figure 2 shows the transmitted ordersof a grating with d ­ 100l. The f ield-stitching andBKK plots are indistinguishable. The plot of the per-centage difference of the field-stitched diffraction ef-ficiencies from the rigorously calculated efficiencies ismore informative. Unsurprisingly, it shows that thelargest discrepancies occur when the values are small-est. In fact, to provide a clearer picture of the accuracywith which the most significant orders are matched, wehave excluded from the comparison all orders with ef-ficiencies of less than 1025. Similarly, in Fig. 3, whichshows a grating with d ­ 10, 000l, the field-stitchingand the scalar results are very close. A small mis-match in the zeroth orders is the noticeable differencefrom the small-period comparison. It should be notedthat the scalar diffraction eff iciencies have been ad-justed by use of the Fresnel ref lection coeff icients toremove ref lected energy.

We have reported results for dielectric gratings intransmission with TE-polarized incident radiation.There is no diff iculty in using the method for TM po-

Fig. 2. Analysis of a 1 3 81 fan-out grating with a pe-riod of 100 wavelengths. The dotted curve shows thediffracted-order efficiencies as calculated with the rigorousBKK method. The f ield-stitching curve is indistinguish-able on this plot. The dashed curve represents the per-centage difference of the f ield-stitched eff iciency from therigorous eff iciency. Field-stitching parameters are w ­10, l ­ 201.

1510 OPTICS LETTERS / Vol. 21, No. 18 / September 15, 1996

Fig. 3. Analysis of a 1 3 81 fan-out grating with a pe-riod of 10,000 wavelengths. The dotted curve shows thediffracted-order efficiencies as calculated with paraxialscalar theory (with Fresnel ref lection adjustment). Thefield-stitching curve is also plotted but is only distin-guishable from the scalar curve in the zeroth order. Thedashed curve represents the percentage difference of thefield-stitched efficiency from the scalar efficiency. Field-stitching parameters are w ­ 20, l ­ 151.

larization, although the choice of the rigorous diffrac-tion method will have more significance for theaccuracy and speed of the local calculations. Simi-larly, f ield stitching the ref lected field should presentno problems. It is less obvious how the locality ofresonance effects in metallic gratings will comparewith that of dielectric gratings. Certainly, one wouldnot expect the exponentially decaying waves in thebody of the grating to result in long-range coupling.In any case, it is intuitively reasonable that the depthof the grating will largely determine the range of theresonance effects and, hence, the necessary size ofthe local period. This is one limit to the accurateapplication of the f ield-stitching method. One couldquantify the extent of the approximation by examiningthe size of the f ield discontinuities at stitching bounda-ries. Furthermore, apart from the accumulation ofrounding errors owing to finite precision of numbers asrepresented on a computer, there is no decrease in theaccuracy of the f ield-stitching method as the numberof sections stitched increases. The algorithm is asaccurate with many sections as with few. This can beseen from the expression for Tp in Eq. (3). Only theaverage error in the total field per unit distance alongthe grating affects the values of the diffracted orders.

There are numerous potential applications of thefield-stitching method in diffractive optics. For

example, during optimization the time taken toevaluate the grating performance is critical, becauseevaluation must be done so often. The relativespeed of f ield stitching compared with that of a fullrigorous method will provide significant improve-ment for some gratings. In addition, if suitablelook-up tables are kept, then a local alteration inthe grating structure requires reevaluation of onlyfew grating sections for the new performance tobe found. This will permit optimization of largerresonance-regime gratings than has previously beenachieved. For example, fan-out to a greater num-ber of orders will be possible. Another interestingpossibility is the analysis of aperiodic structures.Sheridan and Sheppard5 and Montiel and Neviere6

have studied isolated features and linear zone plates,respectively, by evaluating the diffracted f ield of anarray of such elements, using a grating method. Ifthe distance between the elements is large enough,they are effectively decoupled, and a simple integralrepresentation of the aperiodic field can be obtained.Field stitching could naturally be combined with thisapproach. The study of large (cylindrical) diffractivelenses is then feasible. Also, provided that it variesslowly, a nonuniformly intense incident wave can beeasily modeled. We plan to continue research in thisdirection.

In conclusion, we have presented a general methodfor the analysis of diffraction from gratings of arbi-trarily large period, even those containing wavelength-scale features. We have shown that one can obtainhighly accurate results by studying gratings for whichindependent verification is possible. And we suggestthat the application of field stitching to a variety ofareas within diffractive optics promises accurate analy-sis and high-performance designs.

References

1. M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. 71,811 (1981).

2. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams,and J. R. Andrewartha, Opt. Acta 28, 413 (1981).

3. R. Petit, ed., Electromagnetic Theory of Gratings(Springer-Verlag, Berlin, 1980).

4. K. Knop, J. Opt. Soc. Am. 68, 1206 (1978).5. J. T. Sheridan and C. J. R. Sheppard, J. Opt. Soc. Am. A

4, 614 (1993).6. F. Montiel and M. Neviere, J. Opt. Soc. Am. A 12, 2672

(1995).